MapperGraphen

MapperGraphen, or Mapper-Graphen, denotes the graphical output produced by the Mapper algorithm in topological data analysis when applied to a dataset. The Mapper framework seeks a concise, interpretable summary of the shape of data by combining a filter function with a cover of the data’s range and clustering within the preimages of cover elements. The resulting structure is a graph in which nodes correspond to clusters of points within a region of the cover and edges connect nodes whose clusters share data points in overlapping regions.

Construction typically starts with a dataset X embedded in a metric or topological space and a continuous filter function f: X → R^m. A cover of the image f(X) is chosen using overlapping intervals or hypercubes. For each region in the cover, the subset of X mapped into that region is clustered, and each resulting cluster becomes a node. If two clusters arise from overlapping regions and share common data points, an edge is drawn between their nodes. Higher-dimensional simplices can be considered, but the standard Mapper output is a graph.

Interpretation and use: The Mapper graph encodes the data’s connectivity and branching structure, providing a visual and quantitative summary that supports exploratory data analysis, visualization, and hypothesis generation. It is widely applied in biology, image analysis, finance, and other domains. The results depend on choices of the filter, the cover, and the clustering method, which means parameter sensitivity and non-uniqueness are inherent to the approach.

Relation to related concepts: Mapper is related to Reeb graphs and approximates certain topological features of the data. It has several software implementations and continues to be used alongside other topological data analysis tools to study high-dimensional datasets.